Fall 2018

Basel Number Theory Seminar: Fall 2018

Time: 14.15 on 27 Sep, 2018

David Belius (Univ. of Basel)

Extreme values of the Riemann Zeta function, and log-correlatedGaussian fields

I will describe how the Riemann Zeta function on the critical line canbe viewed as a pseudo-random Gaussian field with a correlationfunction with logarithmic growth. Such log-correlated random fieldshave recently attracted considerable interest in probability theory.Fyodorv, Hiary and Keating conjectured several striking results aboutthe extreme values of the Riemann Zeta function based on thisconnection. In this talk I will explain how a certain approximate treestructure in Dirichlet polynomials can be used to prove one of theirconjectures, giving the asymptotics of the maximum of the magnitude ofthe function in a typical interval of length O(1).

Time: 14.15 on 4 Oct, 2018

Dijana Kreso (TU Graz)

On Goormaghtigh's equation

In my talk I will present results that come from a joint work with M. Bennett and A. Gherga from The University of British Columbia. We studied Goormaghtigh's equation:\begin{equation}\label{eq}\frac{x^m-1}{x-1} = \frac{y^n-1}{y-1}, \; \; y>x>1, \; m > n > 2. \end{equation}There are two known solutions $(x, y,m, n)=(2, 5, 5, 3), (2, 90, 13, 3)$ and it is believed that these are the only solutions. It is not known if this equation has finitely or infinitely many solutions, and not even if that is the case if we fix one of the variables. It is known that there are finitely many solutions if we fix any two variables. Moreover, there are effective results in all cases, except when the two fixed variables are the exponents $m$ and $n$. If the fixed $m$ and $n$ additionally satisfy $\gcd(m-1, n-1)>1$, then there is an effective finiteness result. My co-authors and me showed that if $n \geq 3$ is a fixed integer, then there exists an effectively computable constant $c (n)$ such that $\max \{ x, y, m \} < c (n)$ for all $x, y$ and $m$ that satisfy Goormaghtigh's equation with $\gcd(m-1,n-1)>1$. In case $n \in \{ 3, 4, 5 \}$, we solved the equation completely, subject to this non-coprimality condition.

Time: 14.15 on 1 Nov, 2018

Shabnam Akhtari (Univ. of Oregon / MPIM Bonn)

Representation of integers by binary forms

Let $F(x, y)$ be an irreducible binary form with integer coefficients and of degree at least 3. By a well known result of Thue, the equation $F(x, y) = m$ has only finitely many solutions in integers $x$ and $y$. I will discuss some old and new quantitative results on the number of solutions of such equations. I will also talk about my joint work with Manjul Bhargava, where we use these bounds to show that many equations of the shape $F(x, y) = m$ have no solutions.

Time: 14.15 on 8 Nov, 2018

David Masser (Univ. of Basel)

Solving polynomial-exponential equations.

Inspired by Schanuel's Conjecture, Boris Zilber has proposed a ``Nullstellensatz'' (also conjectural) asserting which sorts of polynomial-exponential equations in several variables have a complex solution. Last year Dale Brownawell and I published a proof in the situation which can be regarded as ``typical''. But it does not cover all situations for two variables, some of which involve simply stated problems in one variable like finding complex $z \neq 0$ with $e^z+e^{1/z}=1$. Recently Vincenzo Mantova and I have settled the general case of two variables. We describe our methods -- for example, to solve $$e^z+e^{\root 9 \of {1-z^9}}=1$$ one approach uses theta functions on ${\bf C}^{28}$.

Time: 14.15 on 22 Nov, 2018

Amador Martin-Pizarro (Univ. of Freiburg)

On bounded automorphisms of fields with operators

Lascar showed that the group of automorphisms of the complex field which fix the algebraic closure of the prime field is simple. For this, he first showed that there are no non-trivial bounded automorphisms. An automorphism is bounded if there is a finite set $A$ such that the image of every element b is algebraic over $A$ together with $b$. The same result holds for a "universal" differentially closed field of characteristic zero, where we replace algebraic by differentially algebraic. Together with T. Blossier and C. Hardouin, we provided in https://arxiv.org/abs/1505.03669 a complete classification of bounded automorphisms in various fields equipped with operators, among others, for generic difference fields in all characteristics or for Hasse-Schmidt differential fields in positive characteristic.

Time: 14.15 on 29 Nov, 2018

Ana María Botero (Univ. of Regensburg)

Integrability of b-divisors on toroidal embeddings

Convex geometry has been widely and successfully used to explore the geometry of algebraic varieties. A well-known class of examples comes from the theory of toric varieties, where the combinatorics of a lattice polytope encrypts most of the geometric properties of the corresponding toric variety. In this talk, we will discuss how to use convex analytical methods on so called weakly embedded polyhedral complexes to obtain integrability results of toroidal b-divisors. The latter can be thought of as a limit of toroidal divisors defined over all possible toroidal compactifications of an algebraic variety. We will also mention applications of these results to Arakelov geometry.

Time: 14.15 on 6 Dec, 2018

Berke Topacogullari (Ecole Polytechnique Fédérale de Lausanne)

Titchmarsh's divisor problem for multiplicative functions

The classical Titchmarsh divisor problem is concerned with the mean behaviourof the divisor function on the set of primes shifted by a non-zero constant. Weconsider analogues of this problem where the set of primes is replaced by othersets of multiplicative nature and also discuss further generalizations. This isjoint work with S. Drappeau.

Time: 14.15 on 13 Dec, 2018

Ramon Moreira Nunes (Ecole Polytechnique Fédérale de Lausanne)

The twelfth moment of Dirichlet L-functions to smooth moduli

Heath-Brown proved a strong bound for the twelfth moment ofthe Riemann zeta function which is not sharp but that has the nicefeature of giving the Weyl estimate as a corollary. In this talk wewill take a look at the key points of Heath-Brown's proof and show howone can deduce an analogous result for Dirichlet L-functions withsmooth moduli. Our result relies on bounds for some complicatedexponential sums that we study by appealing to the theory of algebraictrace functions due to Katz and Fouvry-Kowalski-Michel.

Date	Speaker	Title
27 Sep, 2018 14.15	David Belius (Univ. of Basel)	Extreme values of the Riemann Zeta function, and log-correlatedGaussian fields
4 Oct, 2018 14.15	Dijana Kreso (TU Graz)	On Goormaghtigh's equation
1 Nov, 2018 14.15	Shabnam Akhtari (Univ. of Oregon / MPIM Bonn)	Representation of integers by binary forms
8 Nov, 2018 14.15	David Masser (Univ. of Basel)	Solving polynomial-exponential equations.
22 Nov, 2018 14.15	Amador Martin-Pizarro (Univ. of Freiburg)	On bounded automorphisms of fields with operators
29 Nov, 2018 14.15	Ana María Botero (Univ. of Regensburg)	Integrability of b-divisors on toroidal embeddings
6 Dec, 2018 14.15	Berke Topacogullari (Ecole Polytechnique Fédérale de Lausanne)	Titchmarsh's divisor problem for multiplicative functions
13 Dec, 2018 14.15	Ramon Moreira Nunes (Ecole Polytechnique Fédérale de Lausanne)	The twelfth moment of Dirichlet L-functions to smooth moduli